C4 chapter 2 - parametrics

AKA "coordinate geometry in the (x,y) plane"
This chapter should be called parametric equations, since this is what you will learn to work with.

C4 § 2.1

Prior knowledge:

• You should be familiar with creating tables of values to plot graphs from C1 chapter 2

WALT:

• plot the graphs of functions defined in terms of a parameter t by creating tables of values.

Examples:

1. Draw the curve given by the parametric equations x = 2t and y = t² for -3 ≤ t ≤ 3.
   
   Click on the circular buttons next to the table to plot the key points.
   ...Then click on the circular button next to the Cartesian form to reveal the full graph.

C4 Ex 2A

• C4 Exercise 2A worked solutions:
  part done:
  

  C4 Exercise 2A(parametric equations rearranged to Cartesian form).jnt
  

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  C4 Exercise 2A(parametric equations rearranged to Cartesian form).pdf
  

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C4 § 2.2

C4 Ex 2B

• C4 Exercise 2B worked solutions:
  
  

  C4 Exercise 2B (solve problems involving parametric,).jnt
  

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  C4 Exercise 2B (solve problems involving parametric,).pdf
  

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  If you need solutions to questions I haven't yet done, please ask via the sandbox.
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C4 § 2.3

C4 Ex 2C

• C4 Exercise 2C worked solutions:

C4 § 2.4

Prior knowledge: you are expected to be familiar with the chain rule from C3 chapter ?; you must be fluent with definite integration from C2 chapter
WALT integrate functions given in parametric form by using a version of the chain rule

$\displaystyle \int _{x=a} ^{x=b} {y}\, \mathrm{d}x \equiv \int _{t=p} ^{t=q} {y \, \dfrac {\mathrm{d}x}{\mathrm{d}t}} \, \mathrm{d}t$
This is usually a pretty straightforward case of finding dx/dt and multiplying the answer in terms of t by the function y = f(t) then integrating the result.
The bit to watch out for is the change of limits which I have emphasised by x = a changing to t = p, etcetera. You'll need to find the correct values of t to give you the appropriate values of x from the original limits or picture.
Remember: the key issue is that the limits over which you evaluate the integral must be in terms of the final variable in the chain, so when it's ∫(some junk)dx the limits are values of x, and when it's changed to ∫(some new junk)dt the limits must be the right values of t that, using x = g(t) would give you those values of x.

C4 Ex 2D

• C4 Exercise 2D worked solutions:

C4 Ex 2E

• C4 Exercise 2E worked solutions: